Suppose \(\gamma >0\) . By theorem REF , \(a_N = F^{-1}(1- 1/N)= U_F(N)\) , where the second equality follows by eq. (REF ). By corollary 1.2.10 in [1]} \((U_F(x))^{-1} \in RV_{-\gamma }\) . Then by lemma REF  we can write \(1/U_F(x) = x^{-\gamma } L(x)\)  where \(L\)  is a slowly varying function (that depends on \(F\) ). In particular, \(1/a_{N}= 1/U_F(N) = N^{-\gamma } L(N)\)  .
Then at our choice of \(u_{S, \tau , N, T}\) , again using eq. (REF ), it holds that
\(& \dfrac{1}{a_N}\left(F^{-1}\left(1- \frac{1}{N\tau } \frac{\log (T)}{\log (T)+1} \right) - F^{-1}\left(1- \dfrac{1}{N\tau } \right)\right)\\&= \dfrac{1}{a_N}\left(U_F\left(N\tau \dfrac{\log (T)+1}{\log (T)} \right)- U_F(N\tau ) \right)\\&= N^{-\gamma }L(N)\left( \left(N\tau \dfrac{\log (T)+1}{\log (T)} \right)^{\gamma } \dfrac{1}{L\left( N\tau \frac{\log (T)+1}{\log (T)} \right)} - (N\tau )^{\gamma } \dfrac{1}{L(N\tau )} \right)\\& \propto \left( \dfrac{\log (T)+1}{\log (T)} \right)^{\gamma } \dfrac{L(N) }{L\left( N\tau \frac{\log (T)+1}{\log (T)} \right)} - \dfrac{L(N)}{L(N\tau )} \\& \rightarrow 0.\) 
